Let C[0, 1] have inner product [f(x), g(x)] = ∫ f(x)g(x) dx. Find α so that (x + αx2 ) ⊥ (x + 1)

Q: 2. Find the Taylor Polynomial of degree n for f(x) =, centered at %3D 1-x X = 0.

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Q: Find the Taylor polynomial of degree 2 centered at a=1 that approximates f(x)=ln(3x) P2(x)=

A: The general form of Taylor expansion of a function centered at a is, fx=∑n=0∞fnan!x-an The second…

Q: Let fn E D[0, 1]. Suppose that fh g on [0, 1], and lim fn(1/2) exists. For each x E [0, 1], show…

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Q: Find the 5th degree Taylor Polynomial centered at x=0.  sin(2x) ln(1+x)

A: Given that    sin(2x) and ln(1+x) To find 5th degree Taylor polynomial centered at x=0

Q: lim In (z\x2 + y²) P - (5, 5, 1)

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Q: In Exercises 23–30, find Tn centered at x = a for all n. 26. f (x) = e", a =-2

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Q: Find the third degree Taylor polynomial of f(x)= /x centered at x=1.

A: First, let's understand The Taylor series Expansion of any polynomial and the meaning of…

Q: The Wronskian 1. Show that the functions ex, cos x and sin x are linearly independent.

A: As per guidelines, We’ll answer the first question since the exact one wasn’t specified. Please…

Q: Find the coefficient of degree 2 in Taylor polynomial for f(x) = In (1+ rx) about x = = 1.

A: To find the coefficient of degree 2 in Taylor polynomial for the give function- f(x)=ln(1+rx) about…

Q: 6. Let 1 f (x) - cos x + sin(2x), x E [0, 1] 2 Find the least squares polynomial approximation of…

A: Let fx=12cos x-sin2x, x∈0,1. To find quadratic fitting  fx≈P2x=a0+a1x+a2x2 Consider, s0=∫011dx=1

Q: A continuous r.v. F(x1= === 2 × -IN 15 X has CDF X Co OBX C1 18X 16 17 **** 18

A: Given: Fx=0            x<012x        0≤x<1k            x≥1

Q: Determine the fourth Taylor polynomial of f(x) = ln(1 - x)at x = 0, and use it to estimate ln(0.9).

A: Given function is: f(x) = ln(1 - x) The Taylor series expansion of the above function at x = 0 is:…

Q: Prove that for the gamma function [(x) = ftx-¹e-tdt that for n a positive integer then I(n) = (n-1)!

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Q: 5) (a) Show that F = (8-14xy2 + 2ye²x) + (e²x - 14x²y) y is conservative. (b) Find a potential…

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Q: ind the Taylor polynomial of degree 2 centered at a=π that approximates f(x)=cos(5x) P2(x)=

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Q: What is the minimum order of the Taylor polynomial centered at 0 for cos x required to 10-42

A: Given,        cos(-0.35) with an absolute error no greater than 10-4.

Q: Show (in terms of e - 6) that a function f : [2,7] R be defined by f(z) = V² +1 is uniformly…

A: In this question, we need to show the given function f(x) is uniformly continuous.…

Q: 8. Given for C[0,1] the inner-product function (f,g)=[f(x)g(x)dx , find ||f|| if f(x) = x+1.

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Q: Find the definite integral from 3 to 5 of x+2 / (x^2 - 2x) using partial fraction decomposition

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Q: Example 1.7.3 Prove that dx/x is divergent. 1

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Q: Show that the Dirichlet function f defined on [0 , 1] by f(x) = { 1 if x is rational 0 if x is…

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Q: 3. For Z = x²secy,prove that Zxy is equal toZyx

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Q: 2. Find the Taylor Polynomial of degree n for f (x) = 1-x centered at X = 0.

A: Dear students here I'm answering to this problem according to our company guidelines so, here I…

Q: (a) express ux, u y, and uz as func-tions of x, y, and z both by using the Chain Rule and by…

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Q: Evaluate the integral. x² - 81 | V=² x x² - 81 | Vx² x √x². 81 X √x². 81 X x² - 81 X 1² 1² - dx = -…

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Q: Find the nth Taylor polynomial centered at c. f(x) = ln(x), n = 4, c = 6 P4(x) = X

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Q: Find the degree n=4 Taylor polynomial for f(x) = ln (x + 1) at c=1.

A: The degree n=4 Taylor polynomial for f(x) at c is given by f(x) = f(c) + f(1)(c)1!(x-c) +…

Q: Evaluate the given integral applying Chain Rule * S(V25 + x² ) 2x 2 (25+x²)v25+x² -+ c С. 2…

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Q: 11. Prove the absorption law x +xy=x using the other laws in Table 5.

A: To prove: x+xy=x We use the following identities (1) x=x·1 (2) 1+x=1 (3) xy+z=xy+xz    ( This is…

Q: Find the degree  2 Taylor polynomial T2(x) for f(x) = tan x centered at x = π/3

A: First, we will find the general Taylor series of f(x) = tan x centered at x = π/3. We know that…

Q: Find the Taylor polynomial p2 for f(x) = cos(2x) at x = 5.

A: Given that f(x) = cos(2x) at x= π/8  Find the taylor polynomial p2 . Here i attached solution of…

Q: continuous! bx + a 5x lue of o4h is

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Q: 2n Jnsi (x) = " J,(x) – J(4)

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Q: Find the nth Taylor polynomial centered at c. f(x) = x°cos x, n = 2, SCos C = T

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Q: Find the linearization of y = ev7x at x = 16. (Use symbolic notation and fractions where needed.)

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Q: Find T4(x) Taylor polynomial of degree 5 of the function f(x)=cos(x^3) at a=0

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Q: 7. The function (x, y)= (2x* + y* )exp + xy' is homogenous in x and y of degree

A: .Given,f(x,y) = (2x4+y4) ey2x +x y3Put y= xSo we get,f(x,x)= (2x4+x4) ex2x +x x3 f(x,x)= (3x4) e0.5…

Q: 3.2 Let f(x) = Approximate the integral x4 x L f(x) dx using the Trapezoidal rule with n = 4. -2

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Q: let f(x) = cos(x) and x0 = 0. Find the Taylor polynomial of degree N = 4.

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Q: . Let Show that, Hence show that •2T F₁(x) = 2/1/² 0 Fn(2) = eix cos ∞ N=-∞ eix cose eine de in Jn…

A: As per the question we are given the sequence of functions Fn(x) in form of a Fourier integral. And…

Q: Find the nth Taylor polynomial for the function f(x) = x2 cos x, n = 2, centered at c = .

A: Solution:- Find the nth Taylor polynomial for the function f(x) = x2 cos x, n = 2, centered at c = .…

Q: Show that the family of functions {s()}" COS n=0 is mutually orthogonal on -L≤ x ≤ L, with respect…

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Q: show that f(z)=(z-1) cos(1/z+2) has an essential singularity at z=-2

A: Introduction: Laurent series has an important role in determining the singularities. Suppose, a…

Q: 8. Use the Chain Rule to find az/as and az/at. z = (y-2x)³, x = s³t², y=s+t²

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Q: 4. Expand f (x) = 1 around a = 0, to get linear, quadratic and - 1-x cubic approximations.

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Q: Find the nth Taylor polynomial centered at c. f(x) = ln(x), n = 4, c = 6 P4(X) =

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Q: Let c(t) Use the chain rule to compute (foc)'(t). (cos(t), sin(t), t²) and f(x, y, z) = x² + y² +…

A: The objective is to find the derivative.

Q: y = In[(2x – 1)(x + 3)] with Respect to x by using the chain rule.

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